Utility Maximization Problem with Uncertainty and a Jump Setting

TL;DR

This paper addresses a robust utility maximization problem in a jump setting, establishing the existence of an optimal measure and characterizing the value process via a quadratic-exponential BSDE, advancing stochastic control theory.

Contribution

It introduces a novel framework for utility maximization with jumps, proving existence of optimal measures and linking the value process to a quadratic-exponential BSDE.

Findings

Existence of an optimal probability measure for the robust problem.

Characterization of the value process as a unique solution to a quadratic-exponential BSDE.

Extension of utility maximization theory to jump processes with time consistent penalties.

Abstract

We study a robust utility maximization problem in the unbounded case with a general penalty term and information including jumps. We focus on time consistent penalties and we prove that there exists an optimal probability measure solution of the robust problem. Then, we characterize the dynamic value process of our stochastic control problem as the unique solution of a Quadratic-Exponential BSDE.